Fri Mar 12, 2010
60 Evans Hall, 4:10–6 PM
Grigor Sargsyan (UCLA)
Descriptive Inner Model theory
It is by now a well-known theorem of Martin, Steel, and Woodin that if there is a measurable cardinal above omega-many Woodin cardinals then AD holds in L(R). Since this theorem was proved, many theorems of a similar nature have appeared in set theory. For instance, Steel recently showed that PFA, the proper forcing axiom, implies that AD holds in L(R). Many results that followed the Martin-Steel-Woodin theorem, including Steel’s aforementioned result, used a substantial amount of inner model theory to prove that all sets in a certain “canonical” collection of sets of reals are determined. In this talk we will take canonical stand for Universally Baire. Letting then Gamma be the collection of Universally Baire sets, many of the theorems that have been proven after the Martin-Steel-Woodin result are about the size of Gamma, and in particular, the earlier results just show that under certain hypotheses all sets of reals in L(R) are in Gamma. Descriptive inner model theory then can be defined as the study of the properties of Gamma in various mathematically rich extensions of ZFC, using methods from inner model theory. Inner model theory enters into the study of the properties of Gamma through a conjecture known as the Mouse Set Conjecture, which allows one to translate questions about sets of reals into questions about iteration strategies for a certain kind of mice known as hod mice. In this talk, we will explain all this terminology and will outline some of the more recent theorems, including the following.
Theorem (S.) If there is a Woodin limit of Woodins then there is Gamma^* which is a subset of Gamma and such that and L(Gamma^*, R) satisfies AD_R + Theta is regular.
We will also explain some applications in calibrating lower bounds of consistency strength and in the inner model problem.